Tagged Questions
0
votes
0answers
20 views
Supermartingale Lemma + related problems
Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
1
vote
1answer
65 views
Martingale inequality
Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define
$$
Y^r_t := \int_0^t f(r,s) dW_s
$$
For each fixed $r$, ...
0
votes
1answer
183 views
Show that this continuous local martingale is a martingale
We are given the following SDE:
$$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$
and
$$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$
We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
0
votes
0answers
63 views
Local martingale iff each component is a local martingale?
This is probably an easy question:
A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
3
votes
1answer
190 views
$\mathcal{F_t}$-martingales with Itô's formula?
I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar:
...
2
votes
1answer
125 views
Is the solution to a driftless SDE with Lipschitz variation a martingale?
If $\sigma$ is Lipschitz, with Lipschitz constant $K$, and $(X_t)_{t\geq 0}$ solves
$$dX_t=\sigma(X_t)dB_t,$$ where $B$ is a Brownian motion, then is $X$ a martingale? I'm having difficulty getting ...
3
votes
2answers
101 views
If $X$ is a martingale, $X(0)=0$; $f$ left continuous, is $\int f X$ dt also a martingale?
If $X(t)$ is a martingale, and $X(0) = 0$.
$f(t)$ is a left continuous function,
$$
g(t) = \int_0^t f(s) X(s) ds
$$
is $g(t)$ also a martingale?
I guess it shall be, but don't know how to prove ...
0
votes
1answer
228 views
$d$-Dimensional Brownian Motion Martingales
Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a ...
4
votes
1answer
403 views
Martingale problem
If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent:
(i)
for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) ...
4
votes
1answer
530 views
Is this a martingale?
Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution
$$
...
